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CT4: Study Guide

The Actuarial Education Company © IFE: 2016 Examinations

2016 Study Guide

Subject CT4

Introduction This Study Guide contains all the information that you will need before starting to study Subject CT4 for the 2016 exams. Please read this Study Guide carefully before reading the Course Notes, even if you have studied for some actuarial exams before. When studying for the UK actuarial exams, you will need:

a copy of the Formulae and Tables for Examinations of the Faculty of Actuaries and the Institute of Actuaries, 2nd Edition (2002) – these are often referred to as simply the “Yellow Tables”

a “permitted” scientific calculator – you will find the list of permitted calculators on the profession’s website. Please check the list carefully, since it is reviewed each year.

These are both available from the Institute and Faculty of Actuaries’ eShop. Please visit www.actuaries.org.uk.

Contents Section 1 The Subject CT4 course structure Page 2 Section 2 ActEd study support Page 3 Section 3 How to study to pass the exams Page 12 Section 4 Frequently asked questions Page 16 Section 5 Core Reading and the Syllabus Page 17 Section 6 Syllabus Page 21

CT4: Study Guide

© IFE: 2016 Examinations The Actuarial Education Company

1 The Subject CT4 course structure

There are four parts to the Subject CT4 course. The parts cover related topics and have broadly equal lengths. The parts are broken down into chapters. The following table shows how the parts, the chapters and the syllabus items relate to each other. The end columns show how the chapters relate to the days of the regular tutorials. This table should help you plan your progress across the study session.

Part Chapter Title No of pages

Syllabus objectives

2 full days

3 full days

1

1 Principles of actuarial modelling 27 (i)

1

1

2 Stochastic processes 35 (ii)

3 Markov chains 64 (iii)

4 The two-state Markov model 37

(v)8

(vii)1-3

2

5 Time-homogeneous Markov jump processes 61

(iv)1-4

(vii)1-3

6 Time-inhomogeneous Markov jump processes

60 (iv)1, 3-8

2

3

7 Survival models 42 (v)1-7

2

8 Estimating the lifetime distribution function

60 (vi)1-4

9 Proportional hazards models 49 (vi)5-6

3 4

10 The Binomial and Poisson models 32

(vii)4

(viii)

11 Exposed to risk 30 (ix)

12 Graduation and statistical tests 72 (x)1-3

13 Methods of graduation 44 (x)4-7

CT4: Study Guide


2 ActEd study support

Successful students tend to undertake three main study activities:

1. Learning – initial study and understanding of subject material

2. Revision – learning subject material and preparing to tackle exam-style questions

3. Rehearsal – answering exam-style questions, culminating in answering questions at exam speed without notes.

Different approaches suit different people. For example, you may like to learn material gradually over the months running up to the exams or you may do your revision in a shorter period just before the exams. Also, these three activities will almost certainly overlap. We offer a flexible range of products to suit you and let you control your own learning and exam preparation. The following table shows the products that we produce. Note that not all products are available for all subjects.

LEARNING

Course Notes

LEARNING &REVISION

Q&A Bank

X Assignments

Combined

Materials Pack (CMP)

X Assignment

Marking

Tutorials

Online Classroom

REVISION

Flashcards

Sound Revision

REVISION & REHEARSAL

Revision Notes

ASET

Revision Tutorials

REHEARSAL

Mock Exam A

Additional Mock Pack

(AMP)

Mock / AMP Marking

The products and services available for Subject CT4 are described below.

CT4: Study Guide


“Learning” products

Course Notes

The Course Notes will help you develop the basic knowledge and understanding of principles needed to pass the exam. They incorporate the complete Core Reading and include full explanation of all the syllabus objectives, with worked examples and short questions to test your understanding. Each chapter includes the relevant syllabus objectives, a chapter summary and, where appropriate, a page of important formulae or definitions.

“Learning & revision” products

Question and Answer Bank

The Question and Answer Bank provides a comprehensive bank of questions (including some past exam questions) with full solutions and comments. The Question and Answer Bank is divided into five parts. The first four parts include a range of short and long questions to test your understanding of the corresponding part of the Course Notes. Part five consists of 100 marks of exam-style questions. X Assignments

The four Series X Assignments (X1 to X4) cover the material in Parts 1 to 4 respectively. Assignments X1 and X2 are 80-mark tests and should take you two and a half hours to complete. Assignments X3 and X4 are 100-mark tests and should take you three hours to complete. The actual Subject CT4 examination will have a total of 100 marks. Combined Materials Pack (CMP)

The Combined Materials Pack (CMP) comprises the Course Notes, the Question and Answer Bank and the Series X Assignments. The CMP is available in eBook format for viewing on a range of electronic devices. eBooks can be ordered separately or as an addition to paper products. Visit www.ActEd.co.uk for full details about the eBooks that are available, compatibility with different devices, software requirements and printing restrictions.

CT4: Study Guide


CMP Upgrade The purpose of the CMP Upgrade is to enable you to amend last year’s study material to make it suitable for study for this year. In most cases, it lists all significant changes to the Core Reading and ActEd material so that you can manually amend your notes. The upgrade includes replacement pages and additional pages where appropriate. However, if a large proportion of the material has changed significantly, making it inappropriate to include all changes, the upgrade will only outline what has changed. In this case, we recommend that you purchase a replacement CMP (printed copy or eBook) or Course Notes at a significantly reduced price. The CMP Upgrade can be downloaded free of charge from our website at www.ActEd.co.uk. Alternatively, if the upgrade contains a large number of pages, you may prefer to purchase a hard copy from us at a minimal price to cover production and handling costs. A separate upgrade for eBooks is not produced but a significant discount is available for retakers wishing to re-purchase the latest eBook. X Assignment Marking

We are happy to mark your attempts at the X assignments. Marking is not included with the Assignments or the CMP and you need to order it separately. We recommend that you submit your script by email. Your script will be marked electronically and you will be able to download your marked script via a secure link on the internet. Don’t underestimate the benefits of doing and submitting assignments:

Question practice during this phase of your study gives an early focus on the end goal of answering exam-style questions.

You’re incentivised to keep up with your study plan and get a regular, realistic assessment of progress.

Objective, personalised feedback from a high quality marker will highlight areas on which to work and help with exam technique.

In a recent study, we found that students who attempt more than half the assignments have significantly higher pass rates.

CT4: Study Guide


Series Marking Series Marking applies to a specified subject, session and student. If you purchase Series Marking, you will not be able to defer the marking to a future exam sitting or transfer it to a different subject or student. We typically send out full solutions with the Series X Assignments. However, if you order Series Marking at the same time as you order the Series X Assignments, you can choose whether or not to receive a copy of the solutions in advance. If you choose not to receive them with the study material, you will be able to download the solutions via a secure link on the internet when your marked script is returned (or following the final deadline date if you do not submit a script). If you are having your attempts at the assignments marked by ActEd, you should submit your scripts regularly throughout the session, in accordance with the schedule of recommended dates set out in information provided with the assignments. This will help you to pace your study throughout the session and leave an adequate amount of time for revision and question practice. The recommended submission dates are realistic targets for the majority of students. Your scripts will be returned more quickly if you submit them well before the final deadline dates. Any script submitted after the relevant final deadline date will not be marked. It is your responsibility to ensure that we receive scripts in good time. Marking Vouchers Marking Vouchers give the holder the right to submit a script for marking at any time, irrespective of the individual assignment deadlines, study session, subject or person. Marking Vouchers can be used for any assignment. They are valid for four years from the date of purchase and can be refunded at any time up to the expiry date. Although you may submit your script with a Marking Voucher at any time, you will need to adhere to the explicit Marking Voucher deadline dates to ensure that your script is returned before the date of the exam. The deadline dates are provided with the assignments.

CT4: Study Guide


Tutorials

Our tutorials are specifically designed to develop the knowledge that you will acquire from the course material into the higher-level understanding that is needed to pass the exam. We run a range of different tutorials including face-to-face tutorials at various locations, and Live Online tutorials. Full details are set out in our Tuition Bulletin, which is available from the our website at www.ActEd.co.uk. Regular and Block Tutorials In preparation for these tutorials, we expect you to have read the relevant part(s) of the Course Notes before attending the tutorial so that the group can spend time on exam questions and discussion to develop understanding rather than basic bookwork. You can choose one of the following types of tutorial:

Regular Tutorials (two or three days) spread over the session.

A Block Tutorial (two or three consecutive days) held two to eight weeks before the exam.

Online Classroom

The Online Classroom acts as either a valuable add-on or a great alternative to a face-to-face or Live Online tutorial. At the heart of the Online Classroom in each subject is a comprehensive, easily-searched collection of over 100 tutorial units. These are a mix of:

teaching units, helping you to really get to grips with the course material, and

guided questions, enabling you to learn the most efficient ways to answer questions and avoid common exam pitfalls.

The best way to discover the Online Classroom is to see it in action. You can watch a sample of the Online Classroom tutorial units on our website at www.ActEd.co.uk.

CT4: Study Guide


“Revision” products

For most subjects, there is a lot of material to revise. Finding a way to fit revision into your routine as painlessly as possible has got to be a good strategy! Flashcards are an inexpensive option that can provide a massive boost. They can also provide a variation in activities during a study day, and so help you to maintain concentration and effectiveness. Flashcards

Flashcards are a set of A6-sized cards that cover the key points of the subject that most students want to commit to memory. Each flashcard has questions on one side and the answers on the reverse. We recommend that you use the cards actively and test yourself as you go. Flashcards are available in eBook format for viewing on a range of electronic devices. eBooks can be ordered separately or as an addition to paper products. Visit www.ActEd.co.uk for full details about the eBooks that are available, compatibility with different devices, software requirements and printing restrictions.

“Revision & rehearsal” products

Revision Notes

Our Revision Notes have been designed with input from students to help you revise efficiently. They are suitable for first-time sitters who have worked through the ActEd Course Notes or for retakers (who should find them much more useful and challenging than simply reading through the course again). The Revision Notes are a set of six A5 booklets – perfect for revising on the train or tube to work. Each booklet covers one main theme or a set of related topics from the course and includes:

Core Reading with a set of integrated short questions to develop your bookwork knowledge

relevant past exam questions with concise solutions from the last ten years

detailed analysis of key past exam questions (selected for their difficulty), and

other useful revision aids.

CT4: Study Guide


ActEd Solutions with Exam Technique (ASET)

The ActEd Solutions with Exam Technique (ASET) contains our solutions to the previous four years’ exam papers, ie eight papers, plus comment and explanation. In particular it will highlight how questions might have been analysed and interpreted so as to produce a good solution with a wide range of relevant points. This will be valuable in approaching questions in subsequent examinations. A “Mini-ASET” will also be available in the summer session covering the April exam only. Revision Tutorials

Revision Tutorials are intensive one-day face-to-face tutorials or Live Online tutorials in the final run-up to the exam. They give you the opportunity to practise interpreting and answering past exam questions and to raise any outstanding queries with an ActEd tutor. These courses are most suitable if you have previously attended Regular Tutorials or a Block Tutorial in the same subject. Details of how to apply for our tutorials are set out in our Tuition Bulletin, which is available from the our website at www.ActEd.co.uk.

“Rehearsal” products

Mock Exam A

Mock Exam A is a 100-mark mock exam paper and is a realistic test of your exam preparation. It is based on Mock Exam A from last year but it has been updated to reflect any changes to the Syllabus and Core Reading. Additional Mock Pack (AMP)

The Additional Mock Pack (AMP) consists of two further 100-mark mock exam papers – Mock Exam B and Mock Exam C. This is ideal if you are retaking and have already sat Mock Exam A, or if you just want some extra question practice.

CT4: Study Guide


Mock / AMP Marking

We are happy to mark your attempts at Mock Exam A or the mock exams included within the AMP. The same general principles apply as for the X Assignment Marking. In particular:

Mock Exam Marking is available for Mock Exam A and it applies to a specified subject, session and student

Marking Vouchers can be used for Mock Exam A or the mock exams contained within the AMP; please note that attempts at the AMP can only be marked using Marking Vouchers.

Recall that:

marking is not included with the products themselves and you need to order it separately

we recommend that you submit your script by email

your script will be marked electronically and you will be able to download your marked script via a secure link on the internet.

Queries and feedback

From time to time you may come across something in the study material that is unclear to you. The easiest way to solve such problems is often through discussion with friends, colleagues and peers – they will probably have had similar experiences whilst studying. If there’s no-one at work to talk to then use our discussion forum at www.ActEd.co.uk/forums (or use the link from our home page at www.ActEd.co.uk). Our online forum is dedicated to actuarial students so that you can get help from fellow students on any aspect of your studies from technical issues to study advice. You could also use it to get ideas for revision or for further reading around the subject that you are studying. ActEd tutors will visit the site from time to time to ensure that you are not being led astray and we also post other frequently asked questions from students on the forum as they arise. If you are still stuck, then you can send queries by email to [email protected] (but we recommend that you try the forum first). We will endeavour to contact you as soon as possible after receiving your query but you should be aware that it may take some time to reply to queries, particularly when tutors are away from the office running tutorials. At the busiest teaching times of year, it may take us more than a week to get back to you.

CT4: Study Guide


If you have many queries on the course material, you should raise them at a tutorial or book a personal tuition session with an ActEd tutor. Information about personal tuition is set out in our current brochure. Please email [email protected] for more details. If you find an error in the course, please check the corrections page of our website (www.ActEd.co.uk/Html/paper_corrections.htm) to see if the correction has already been dealt with. Otherwise please send details via email to [email protected] or send a fax to 01235 550085. Each year our tutors work hard to improve the quality of the study material and to ensure that the courses are as clear as possible and free from errors. We are always happy to receive feedback from students, particularly details concerning any errors, contradictions or unclear statements in the courses. If you have any comments on this course please email them to [email protected] or fax them to 01235 550085. Our tutors also work with the profession to suggest developments and improvements to the Syllabus and Core Reading. If you have any comments or concerns about the Syllabus or Core Reading, these can be passed on via ActEd. Alternatively, you can send them directly to the Institute and Faculty of Actuaries’ Examination Team by email to [email protected].

CT4: Study Guide


3 How to study to pass the exams

The CT Subject exams

The Core Reading and exam papers for these subjects tend to be very technical. The exams themselves have many calculation and manipulation questions. The emphasis in the exam will therefore be on understanding the mathematical techniques and applying them to various, frequently unfamiliar, situations. It is important to have a feel for what the numerical answer should be by having a deep understanding of the material and by doing reasonableness checks. Subjects CT2 and CT7 are more “wordy” than the other subjects, including an “essay-style” question or two in Subject CT7. As a high level of mathematics is required in the courses it is important that your mathematical skills are extremely good. If you are a little rusty you may wish to consider buying the Foundation ActEd Course (FAC). This covers all of the mathematical techniques that are required for the CT Subjects, some of which are beyond A-Level (or Higher) standard. It is a reference document to which you can refer when you need help on a particular topic. You will have sat many exams before and will have mastered the exam and revision techniques that suit you. However it is important to note that due to the high volume of work involved in the CT Subjects it is not possible to leave all your revision to the last minute. Students who prepare well in advance have a better chance of passing their exams on the first sitting. Unprepared students find that they are under time pressure in the exam. Therefore it is important to find ways of maximising your score in the shortest possible time. Part of your preparation should be to practise a large number of exam-style questions under timed exam conditions as soon as possible. This will:

help you to develop the necessary understanding of the techniques required

highlight the key topics, which crop up regularly in many different contexts and questions

help you to practise the specific skills that you will need to pass the exam. There are many sources of exam-style questions. You can use past exam papers, the Question and Answer Bank (which includes many past exam questions), assignments, mock exams, the Revision Notes and ASET.

CT4: Study Guide


Overall study plan

We suggest that you develop a realistic study plan, building in time for relaxation and allowing some time for contingencies. Be aware of busy times at work, when you may not be able to take as much study leave as you would like. Once you have set your plan, be determined to stick to it. You don’t have to be too prescriptive at this stage about what precisely you do on each study day. The main thing is to be clear that you will cover all the important activities in an appropriate manner and leave plenty of time for revision and question practice. Aim to manage your study so as to allow plenty of time for the concepts you meet in this course to “bed down” in your mind. Most successful students will probably aim to complete the course at least a month before the exam, thereby leaving a sufficient amount of time for revision. By finishing the course as quickly as possible, you will have a much clearer view of the big picture. It will also allow you to structure your revision so that you can concentrate on the important and difficult areas of the course. A sample CT subject study plan is available on our website at:

www.ActEd.co.uk/html/help_and_advice_study_plans.htm It includes details of useful dates, including assignment deadlines and tutorial finalisation dates.

Study sessions

Only do activities that will increase your chance of passing. Try to avoid including activities for the sake of it and don’t spend time reviewing material that you already understand. You will only improve your chances of passing the exam by getting on top of the material that you currently find difficult. Ideally, each study session should have a specific purpose and be based on a specific task, eg “Finish reading Chapter 3 and attempt Questions 1.4, 1.7 and 1.12 from the Question and Answer Bank”, as opposed to a specific amount of time, eg “Three hours studying the material in Chapter 3”. Try to study somewhere quiet and free from distractions (eg a library or a desk at home dedicated to study). Find out when you operate at your peak, and endeavour to study at those times of the day. This might be between 8am and 10am or could be in the evening. Take short breaks during your study to remain focused – it’s definitely time for a short break if you find that your brain is tired and that your concentration has started to drift from the information in front of you.

CT4: Study Guide


Order of study

We suggest that you work through each of the chapters in turn. To get the maximum benefit from each chapter you should proceed in the following order: 1. Read the Syllabus Objectives. These are set out in the box on page 1 of each

chapter. 2. Read the Chapter Summary at the end of each chapter. This will give you a useful

overview of the material that you are about to study and help you to appreciate the context of the ideas that you meet.

3. Study the Course Notes in detail, annotating them and possibly making your own

notes. Try the self-assessment questions as you come to them. Our suggested solutions are at the end of each chapter. As you study, pay particular attention to the listing of the Syllabus Objectives and to the Core Reading.

4. Read the Chapter Summary again carefully. If there are any ideas that you can’t

remember covering in the Course Notes, read the relevant section of the notes again to refresh your memory.

It’s a fact that people are more likely to remember something if they review it several times. So, do look over the chapters you have studied so far from time to time. It is useful to re-read the Chapter Summaries or to try the self-assessment questions again a few days after reading the chapter itself. You may like to attempt some questions from the Question and Answer Bank when you have completed a part of the course. It’s a good idea to annotate the questions with details of when you attempted each one. This makes it easier to ensure that you try all of the questions as part of your revision without repeating any that you got right first time. Once you’ve read the relevant part of the notes and tried a selection of questions from the Question and Answer Bank (and attended a tutorial, if appropriate) you should attempt the corresponding assignment. If you submit your assignment for marking, spend some time looking through it carefully when it is returned. It can seem a bit depressing to analyse the errors you made, but you will increase your chances of passing the exam by learning from your mistakes. The markers will try their best to provide practical comments to help you to improve. To be really prepared for the exam, you should not only know and understand the Core Reading but also be aware of what the examiners will expect. Your revision programme should include plenty of question practice so that you are aware of the typical style, content and marking structure of exam questions. You should attempt as many questions as you can from the Question and Answer Bank and past exam papers.

CT4: Study Guide


Active study

Here are some techniques that may help you to study actively. 1. Don’t believe everything you read! Good students tend to question everything

that they read. They will ask “why, how, what for, when?” when confronted with a new concept, and they will apply their own judgement. This contrasts with those who unquestioningly believe what they are told, learn it thoroughly, and reproduce it (unquestioningly?) in response to exam questions.

2. Another useful technique as you read the Course Notes is to think of possible

questions that the examiners could ask. This will help you to understand the examiners’ point of view and should mean that there are fewer nasty surprises in the exam room! Use the Syllabus to help you make up questions.

3. Annotate your notes with your own ideas and questions. This will make you

study more actively and will help when you come to review and revise the material. Do not simply copy out the notes without thinking about the issues.

4. Attempt the questions in the notes as you work through the course. Write down

your answer before you refer to the solution. 5. Attempt other questions and assignments on a similar basis, ie write down your

answer before looking at the solution provided. Attempting the assignments under exam conditions has some particular benefits:

It forces you to think and act in a way that is similar to how you will behave in the exam.

When you have your assignments marked it is much more useful if the marker’s comments can show you how to improve your performance under exam conditions than your performance when you have access to the notes and are under no time pressure.

The knowledge that you are going to do an assignment under exam conditions and then submit it (however good or bad) for marking can act as a powerful incentive to make you study each part as well as possible.

It is also quicker than trying to write perfect answers. 6. Sit a mock exam four to six weeks before the real exam to identify your

weaknesses and work to improve them. You could use a mock exam written by ActEd or a past exam paper.

CT4: Study Guide


4 Frequently asked questions

Q: What knowledge of earlier subjects should I have? A: The Course Notes are written on the assumption that students have studied

Subject CT3. Q: What level of mathematics is required? A: The level of maths you need for this course is broadly A-level standard.

However, there may be some symbols (eg the gamma function) that are not usually included on A-level syllabuses. You will find the course (and the exam!) much easier if you feel comfortable with the mathematical techniques used in the course and you feel confident in applying them yourself. If you feel that you need to brush up on your mathematical skills before starting the course, you may find it useful to study the Foundation ActEd Course (FAC) or read an appropriate textbook. The full Syllabus for FAC, a sample of the Course Notes and an Initial Assessment to test your mathematical skills can be found on our website at www.ActEd.co.uk.

Q: What calculators am I allowed to use in the exam? A: Please refer to www.actuaries.org.uk for the latest advice.

CT4: Study Guide


5 Core Reading and the Syllabus

Core Reading

The Syllabus for Subject CT4, and the Core Reading that supplements it, has been produced by the Institute and Faculty of Actuaries. The relevant individual Syllabus Objectives are included at the start of each course chapter and a complete copy of the Syllabus is included in Section 6 of this Study Guide. We recommend that you use the Syllabus as an important part of your study. The purpose of the Core Reading is to assist in ensuring that tutors, students and examiners have a clear, shared appreciation of the requirements of the Syllabus. The Core Reading supports coverage of the Syllabus in helping to ensure that both depth and breadth are re-enforced. It is therefore important that students have a good understanding of the concepts covered by the Core Reading. Core Reading deals with each syllabus objective. Core Reading covers what is needed to pass the exam but the tuition material that has been written by ActEd enhances it by giving examples and further explanation of key points. The Subject CT4 Course Notes include the Core Reading in full, integrated throughout the course. Here is an excerpt from some ActEd Course Notes to show you how to identify Core Reading and the ActEd material. Core Reading is shown in this bold font.

Note that in the example given above, the index will fall if the actual share price goes below the theoretical ex-rights share price. Again, this is consistent with what would happen to an underlying portfolio. After allowing for chain-linking, the formula for the investment index then becomes:

I t

N P

B t

i t i ti( )

( )

, ,

where Ni t, is the number of shares issued for the ith constituent at time t; B t( ) is the base value, or divisor, at time t.

This is Core Reading

This is ActEd text

CT4: Study Guide


Core Reading accreditation

The Institute and Faculty of Actuaries would like to thank the numerous people who have helped in the development of this material and in the previous versions of Core Reading. The following paper has been used as the basis for several Units: Macdonald A S, An Actuarial Survey of Statistical Models for Decrement and Transition Data, British Actuarial Journal 2 (1996), by permission of the Editor of the British Actuarial Journal.

Changes to the Syllabus and Core Reading

The Syllabus and Core Reading are updated as at 31 May each year. The exams in April and September / October 2016 will be based on the Syllabus and Core Reading as at 31 May 2015. We recommend that you always use the up-to-date Core Reading to prepare for the exams.

The Institute and Faculty of Actuaries’ Copyright

All study material produced by ActEd is copyright and is sold for the exclusive use of the purchaser. The copyright is owned by Institute and Faculty Education Limited, a subsidiary of the Institute and Faculty of Actuaries. Unless prior authority is granted by ActEd, you may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of the study material. You must take care of your study material to ensure that it is not used or copied by anybody else. Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the Institute and Faculty of Actuaries or through your employer. These conditions remain in force after you have finished using the course.

Past exam papers

You can download some past exam papers and Examiners’ Reports from the profession’s website at www.actuaries.org.uk.

CT4: Study Guide


Further reading

The exam will be based on the relevant Syllabus and Core Reading and the ActEd course material will be the main source of tuition for students. However, some students may find it useful to obtain a different viewpoint on a particular topic covered in Subject CT4. The following list of further reading for Subject CT4 has been prepared by the Institute and Faculty of Actuaries. This list is not exhaustive and other useful material may be available. Basic stochastic processes; A course through exercises. – Brzezniak, Zdzislaw; Zastawniak, Tomasz.– Springer, 1998. – x, 225 pages. – ISBN: 3 540 76175 6. Available from the Publications Unit. Introduction to actuarial modeling. – Hickman, James C. North American Actuarial Journal (1997) 1(3) 1-5. URL: http://www.soa.org/bookstore/naaj_archive.html Modeling, analysis, design, and control of stochastic systems. – Kulkarni, Vidyadhar G. – Springer, 1999. – xiv, 374 pages. – ISBN: 0 387 98725 8. Probability and random processes. – Grimmett, Geoffrey; Stirzaker, David. – 3rd ed. – Oxford University Press, 2001. – xii, 596 pages. – ISBN: 0 19 857222 0. Actuarial mathematics. - Bowers, Newton L; Gerber, Hans U; Hickman, James C; Jones, Donald A; Nesbitt, Cecil J. - 2nd ed. - Society of Actuaries, 1997. - xxvi, 753 pages. - ISBN: 0 938959 46 8. Actuarial models for disability insurance. – Haberman, Steven; Pitacco, Ermanno .–Chapman & Hall, 1999. – xviii, 280 pages. – ISBN: 0 8493 0389 3. Available from the Publications Unit. Analysing survival data from clinical trials and observational studies. – Marubini, Ettore; Valsecchi, Maria Grazia. – John Wiley, 1995. – xvi, 414 pages. – ISBN: 0 471 93987 0. Life contingencies. – Neill, Alistair. – Heinemann, 1977. – vii, 452 pages. – ISBN: 0 434 91440 1. (This book is no longer in print, but has been used as a textbook under earlier education strategies. You should find it relatively easy to borrow a copy from a colleague. Alternatively you can borrow it from the libraries.)

CT4: Study Guide


Life insurance mathematics. – Gerber, Hans U. – 3rd ed. – Springer. Swiss Association of Actuaries, 1997. – 217 pages. – ISBN: 3 540 62242 X. Available from the Publications Unit. Mortality studies. – Scott, William F. – Department of Mathematical Sciences, University of Aberdeen, 2000. – 147 pages. Available from the Publications Unit. Survival models and data analysis. – Elandt-Johnson, Regina C; Johnson, Norman L. – Classics Library ed. – John Wiley & Sons, 1999. – xvi, 457 pages. – ISBN: 0 471 34992 5. Available from the Publications Unit.

CT4: Study Guide


6 Syllabus

The full Syllabus for Subject CT4 is given here. To the right of each objective are the chapter numbers in which the objective is covered in the ActEd course. Aim The aim of the Models subject is to provide a grounding in stochastic processes and survival models and their application. Links to other subjects Subject CT1 — Financial Mathematics: provides an introduction to stochastic interest rates. Subject CT3 — Probability and Mathematical Statistics: introduces the concepts of statistical distributions and modelling. Subject CT5 — Contingencies develops the application of Markov chains. Subject CT8 — Financial Economics: develops the concepts introduced here further. Subject CA1 — Actuarial Risk Management and the Specialist Technical subjects use the models and principles introduced in this subject. Objectives On completion of the subject the trainee actuary will be able to: (i) Describe the principles of actuarial modelling. (Chapter 1)

1. Describe why and how models are used.

2. Explain the benefits and limitations of modelling. 3. Explain the difference between a stochastic and a deterministic model,

and identify the advantages/disadvantages of each. 4. Describe, in general terms, how to decide whether a model is suitable for

any particular application.

CT4: Study Guide


5. Explain the difference between the short-run and long-run properties of a model, and how this may be relevant in deciding whether a model is suitable for any particular application.

6. Describe, in general terms, how to analyse the potential output from a

model, and explain why this is relevant to the choice of model.

7. Describe the process of sensitivity testing of assumptions and explain why this forms an important part of the modelling process.

8. Explain the factors that must be considered when communicating the

results following the application of a model. (ii) Describe the general principles of stochastic processes, and their classification

into different types. (Chapter 2)

1. Define in general terms a stochastic process and in particular a counting process.

2. Classify a stochastic process according to whether it:

operates in continuous or discrete time

has a continuous or a discrete state space

is a mixed type

and give examples of each type of process.

3. Describe possible applications of mixed processes. 4. Explain what is meant by the Markov property in the context of a

stochastic process and in terms of filtrations.

(iii) Define and apply a Markov chain. (Chapter 3)

1. State the essential features of a Markov chain model. 2. State the Chapman-Kolmogorov equations that represent a Markov chain. 3. Calculate the stationary distribution for a Markov chain in simple cases. 4. Describe a system of frequency based experience rating in terms of a

Markov chain and describe other simple applications.

CT4: Study Guide


5. Describe a time-inhomogeneous Markov chain model and describe simple applications.

6. Demonstrate how Markov chains can be used as a tool for modelling and

how they can be simulated. (iv) Define and apply a Markov process. (Chapters 5 and 6)

1. State the essential features of a Markov process model. 2. Define a Poisson process, derive the distribution of the number of events

in a given time interval, derive the distribution of inter-event times, and apply these results.

3. Derive the Kolmogorov equations for a Markov process with time

independent and time/age dependent transition intensities. 4. Solve the Kolmogorov equations in simple cases. 5. Describe simple survival models, sickness models and marriage models

in terms of Markov processes and describe other simple applications. 6. State the Kolmogorov equations for a model where the transition

intensities depend not only on age/time, but also on the duration of stay in one or more states.

7. Describe sickness and marriage models in terms of duration dependent

Markov processes and describe other simple applications.

8. Demonstrate how Markov jump processes can be used as a tool for modelling and how they can be simulated.

(v) Explain the concept of survival models.

1. Describe the model of lifetime or failure time from age x as a random variable. (Chapter 7)

2. State the consistency condition between the random variable representing

lifetimes from different ages. (Chapter 7) 3. Define the distribution and density functions of the random future

lifetime, the survival function, the force of mortality or hazard rate, and derive relationships between them. (Chapter 7)

CT4: Study Guide


4. Define the actuarial symbols t xp and t xq and derive integral formulae

for them. (Chapter 7) 5. State the Gompertz and Makeham laws of mortality. (Chapter 7) 6. Define the curtate future lifetime from age x and state its probability

function. (Chapter 7) 7. Define the expected value and variance of the complete and curtate future

lifetimes and derive expressions for them. Define the symbols xe and xe

and derive an approximate relation between them. (Chapter 7) 8. Describe the two-state model of a single decrement and compare its

assumptions with those of the random lifetime model. (Chapter 4) (vi) Describe estimation procedures for lifetime distributions.

1. Describe the various ways in which lifetime data might be censored. (Chapter 8) 2. Describe the estimation of the empirical survival function in the absence

of censoring, and what problems are introduced by censoring. (Chapter 8) 3. Describe the Kaplan-Meier (or product limit) estimator of the survival

function in the presence of censoring, compute it from typical data and estimate its variance. (Chapter 8)

4. Describe the Nelson-Aalen estimator of the cumulative hazard rate in the

presence of censoring, compute it from typical data and estimate its variance. (Chapter 8)

5. Describe models for proportional hazards, and how these models can be

used to estimate the impact of covariates on the hazard. (Chapter 9)

6. Describe the Cox model for proportional hazards, derive the partial likelihood estimate in the absence of ties, and state the asymptotic distribution of the partial likelihood estimator. (Chapter 9)

(vii) Derive maximum likelihood estimators for the transition intensities in models of

transfers between states with piecewise constant transition intensities.

1. Describe an observational plan in respect of a finite number of individuals observed during a finite period of time, and define the resulting statistics, including the waiting times. (Chapters 4 and 5)

CT4: Study Guide


2. Derive the likelihood function for constant transition intensities in a Markov model of transfers between states given the statistics in 1.

(Chapters 4 and 5) 3. Derive maximum likelihood estimators for the transition intensities in 2.

and state their asymptotic joint distribution. (Chapters 4 and 5) 4. State the Poisson approximation to the estimator in 3. in the case of a

single decrement. (Chapter 10) (viii) Describe the Binomial model of mortality, derive a maximum likelihood

estimator for the probability of death and compare the Binomial model with the multiple state models. (Chapter 10)

1. Describe the Binomial model of the mortality of a group of identical

individuals subject to no other decrements between two given ages, in the naive case where all lives are observed for exactly one year.

2. Describe the advantages of the multiple state Markov model compared

with the Binomial and Poisson models, in terms of:

(a) how well each model represents the process being modelled

(b) how easy it is to find, characterise and use the model parameters, given the data, and

(c) how easily each model is extended to problems other than the study of human mortality.

(ix) Describe how to estimate transition intensities depending on age, exactly or

using the census approximation. (Chapter 11)

1. Explain the importance of dividing the data into homogeneous classes, including subdivision by age and sex.

2. Describe the principle of correspondence and explain its fundamental

importance in the estimation procedure. 3. Specify the data needed for the exact calculation of a central exposed to

risk (waiting time) depending on age and sex. 4. Calculate a central exposed to risk given the data in 3. 5. Explain how to obtain estimates of transition probabilities, including in

the single decrement model the actuarial estimate based on the simple adjustment to the central exposed to risk.

CT4: Study Guide


6. Explain the assumptions underlying the census approximation of waiting times.

7. Explain the concept of rate interval.

8. Develop census formulae given age at birthday where the age may be classified as next, last, or nearest relative to the birthday as appropriate.

The deaths and census data may use different definitions of age. 9. Specify the age to which estimates of transition intensities or

probabilities in 8 apply. (x) Describe how to test crude estimates for consistency with a standard table or a

set of graduated estimates, and describe the process of graduation.

1. Describe the following statistical tests of crude estimates, for comparison with a standard table: (Chapter 12)

chi-square test

standardised deviations test

sign test

cumulative deviation test

grouping of signs test

serial correlations test For each test describe:

the formulation of the hypothesis the test statistic the distribution of the test statistic using approximations where

appropriate the application of the test statistic

2. Describe the reasons for graduating crude estimates of transition

intensities or probabilities, and state the desirable properties of a set of graduated estimates. (Chapter 12)

3. Describe a test for smoothness of a set of graduated estimates. (Chapter 12)

CT4: Study Guide


4. Describe the process of graduation by the following methods, and state the advantages and disadvantages of each: (Chapter 13)

parametric formula standard table graphical

(The student will not be required to carry out a graduation.)

5. Describe how the tests in 1. should be amended to compare crude and graduated sets of estimates. (Chapter 13)

6. Describe how the tests in 1. should be amended to allow for the presence of duplicate policies. (Chapter 13)

7. Carry out a comparison of a set of crude estimates and a standard table,

or of a set of crude estimates and a set of graduated estimates. (Chapter 13)

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